Origen
October 14, 2019, 9:40am
1
I’ve been reading about mental calculus for a while, but I’ve recently realized that I’ve never found anything teaching how to do mental division without resorting to the traditional method. Do you know any method of division? Any different from the traditional one used in schools?
bjoern.gumboldt
October 14, 2019, 10:33am
2
If your unit digit is 9, you can use the tenths digit + 1 and simply carry over the remainder.
12/19 for example… use 2 as 19+1=20
12/2=6
So far you got: .631 now put the remainder to the left of the last digit you calculated to get 11
11/2=5 remainder 1
Now you got: .6315 again you put the remainder left to get 15
15/2=7 remainder 1
Now you got: .63157 put the remainder left to get 17
17/2=8 remainder 1
18/2=9
9/2=4 remainder 1
Now you got: .63157894 and again the remainder to the left to get 14
14/2=7 which is the same as denominator (19) - numerator (12) i.e. 7 and since we dividing by a prime number you can now simply use the 9 compliments to fill the rest.
631578947
368421052
999999999
12/19= .631578947368421052 (repeating)
If you had 15/29 you’d use 3
15/3=5
Next one is 1 for so far: .517241 and again you can stop at 29-15=14 because 29 is also prime and fill up with 9
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Origen
October 14, 2019, 5:04pm
3
Very interesting this method!
bjoern.gumboldt
October 14, 2019, 5:55pm
4
You could also do it right-to-left instead of left-to-right and essentially turn it into a multiplication.
I prefer multiplication but I’m pretty much doing the same thing that @Kinma is doing backwards. Also, since 59 is a prime, it only has 58 recurring decimals, so @albinoblanke could have just continued writing down that number because the last 8 is the same as the first 8.
…also note that if I cut his sequence in half so it’s easier to so, the digits of the top and bottom row always add to 9. So you only have to do half the work (the bottom row) and then add the difference from 9 on the top …
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You might want to google vedic division.
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Kinma
October 15, 2019, 12:21pm
7
I have written a lot on this forum about mental division.
In two threads I showed how to make a division simpler by changing the divisor to a round number that is easier to work with:
Now let’s take a division by 23. We can either move 23 up to 25, or we can change it down to 20. Let’s do both and see which one is easier.
Btw., I consider 25 a ‘round’ number in terms of division. ‘Round’ being easy to use. Division by 25 is the same as multiplication by 4.
First moving up to 25.
The difference 25-23 is 2 so we correct the remainder by a factor of…
When mentally dividing, sometimes it makes sense - speed wise - to do 2 digits at the same time.
An example will make this clear, I hope.
100 / 37
If we were to divide by 40 as in 100 / 40 we immediately see 25 X 40 = 1000.
So divide 1000 by 40 = 25. Remainder is 0.
Then realise, since we are dividing by 37 instead of 40, the remainder is actually 3 x 25 = 75.
This remainder of 75 can be divided by 37. 75 / 37 = 2 with a remainder of 1.
Add the 2 to the 25 to get 27.
Realise this remaind…
Let’s do 100/13 and forget about the digital separator for now. We focus on the digits since we know where the separator must come, right?
If you are new to doing division with negative remainders, see here: Making two digit (or more) mental division easier by using negative remainders
100 goes about 7 times into 13. Let’s do the 2 step process again:
100 - 70 = 30. 30 - 21 = 9
Answer so far: 7 R 9
9 → 90. 90 goes about 7 times into 13:
90 - 70 = 20. 20 - 21 = -1
Answer so far: 77 R -1
…
Has anyone built up a nice repository of how to think about division online or in the books? At this point I have mostly just attempted to improve my multiplication skills in order to better handle this but would appreciate any good reference material to study or work through.
Thanks,
R.
We had some fun analyzing Rüdiger Gamms performance in his division by 109.
Now let’s see how one can easily do the division by 167 in this video (starts at 0:00):
[ - YouTube]
If you did not watch the video, he calculates: 62/167.
I have an idea how to do this calculation with ease, but first I would like to hear from you how you would do this calculation.
The same trick we did with division by numbers ending in 8 or 9 can be done with numbers ending in 1 or 2.
However; instead of adding numbers to the remainder we now subtract numbers from it.
An example will make this clear.
100/61:
We start with dividing 10 by 6:
10/6=1r4. We now subtract the digit found (1) from the remainder. 4 becomes 40, from which we subtract 1 to get 39.
We continue with the remainder of 39. 39/6=6r3. the remainder of 3 becomes 30 and from 30 we subtract 6 to get 24 …
While watching YouTube videos’of Rüdiger Gamms performance in order to find out how he does the higher powers I realized something.
In at least three video’s I saw him do a division by 109. There is a reason for this.
One example is this video:
[Extraordinary People Rüdiger Gamm.flv]
The calculation starts at 0:50.
Watch the video first before reading how this is done.
So first I will show how to do this calculation mentally.
If it looks difficult, dividing by 109, consider div…
I have discussed a similar topic with Kinma but I’ll post here to perhaps broaden the discussion. I find that one of the most common mental calculations I need to perform is where I need to divide by 7.25, 4.2, 72, 58. I realise for 7.25 I could use 7 but I want a little more accuracy, the answer only needs to be accurate to 1 decimal place maybe 2. What is important though is that I produce an answer quickly, so am happy to sacrifice some accuracy for speed. I guess I’m looking for a way to imp…
We haven’t talked about division lately, so let’s revisit.
The other day; I was buying flowers and small talking the seller about the flowers.
He was telling me that for him the roses where 1.85 euros to buy and that he actually needs to sell them for 6 euros (ouch btw), but because he thought that was too expensive he sold them for 5 euros.
I was curious about the margin he calculates with, so I needed to calculate 6/1.85.
As you know if you read my previous posts, I usually start with a ba…
Three digit division.
For most people this is a challenge.
Sometimes there are shortcuts, like the ones we discussed here:
In other cases you need a general way of working that is suitable for mental calculation.
I present here my general way of working.
It involves mostly 2 digit subtractions with each step. Or 3 digits where the 3rd digit is a zero. So each step should be easy to do.
The hard part - as always - is focusing on the numbers and keeping the numbers in your head.
An example…
There is a simple trick to quickly do this.
As an example let’s take 100/59.
In order to calculate this quickly., instead of 100/59, we do 10/6 with a small and simple correction at each step. We add - concatenate might the better word - the digit found to the remainder.
Instead of 100/59, we start with 10/6:
10/6=1r4. Concatenate the ‘1’ to the remainder. So 1r4 becomes 1r41.
41/6=6r5 => 6r56
56/6=9r2 => 9r29
29/6=4r5 => 4r54
54/6=9r0 => 9r09
9/6=1r3 => 1r31
31/6=5r1 => 5r15
15/6=2r3…
In a chat with Tiger, the following came up:
Work with both positive and negative remainders.
Why would you do this? Well, it keeps the digits you work with down to 0-5, which is easier than to work with 0-9.
What I mean is, that for example if you need to divide by 42, you have probably learned that if the next digit is 9 you would first need to work out 42X9 and then subtract. While this is a consistent system and doable on paper, this is difficult to do in your head.
In my way of calculat…
While doing the leapfrog division for 13/97 it occurred to me that in this case using two digit division produces digits a lot faster.
This does not mean that there is something wrong with the leapfrog division. There isn’t. It just means that there are multiple ways of dividing and sometimes one is quicker and other times the other is quicker.
The procedure is dividing by 100 and then adding 3 times the digits produced to the remainder. This is the standard correction needed to go from dividi…
Many of us interested in mental calculation are familiar with the trick for dividing the numbers 1 through 6 by 7. You first memorize the sequence 142857. Depending on the number you’re dividing, you choose a new starting point, and repeat the sequence endlessly from there. For example, 4 divided by 7 is 0.57142857142857142857… and so on.
Numbers that have this quality are known as full reptend primes , and 7 isn’t the only one. The numbers 17, 19, 23, 29, 47, and more all work this same way. Th…
How to mentally calculate 52/59?
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